Prove That ABCD Is a Parallelogram

In geometry, proving that a quadrilateral is a parallelogram can be a challenging task. One way to do this is by using various theorems and properties of triangles and quadrilaterals. In this article, we will prove that ABCD is a parallelogram.

To begin with, let's recall some fundamental concepts. A quadrilateral is said to be cyclic if its vertices lie on a circle. On the other hand, a parallelogram is a quadrilateral with opposite sides of equal length and parallel to each other.

Now, let's consider the given figure ABCD, where AD = BC. We are required to prove that P, Q, C, and D are concyclic. In other words, we need to show that these four points lie on a circle.

The first step is to draw a circle through A and B such that it intersects AD at P and BC at Q. Once this is done, we can use the exterior angle property of a cyclic quadrilateral to our advantage.

Recall that the sum of opposite angles in a cyclic quadrilateral is equal to 180 degrees. This means that ∠APD + ∠PDC = 180 degrees. Now, since P and Q are the points of intersection of the circle with AD and BC respectively, we can write:

∠APD = ∠PQC

Substituting this into our previous equation, we get:

∠PQC + ∠PDC = 180 degrees

Since PQCD is a parallelogram (AD = BC), the opposite angles PQC and PDC are equal. Therefore:

∠PQC = ∠PDC

This implies that ∠APD = ∠PDC as well.

Now, let's use the interior angle property of a triangle to our advantage. Recall that the sum of interior angles on the same side of a triangle is 180 degrees. Applying this to △APC, we get:

∠APC + ∠CPD + ∠PAC = 180 degrees

But ∠APC and ∠PAC are equal (since they are opposite angles of a parallelogram), so we can simplify the equation as follows:

2∠CPD = 180 degrees

This implies that ∠CPD = 90 degrees. Since P, Q, C, and D lie on a circle, this means that P, Q, C, and D are concyclic.

In conclusion, we have proved that ABCD is a parallelogram by showing that P, Q, C, and D are concyclic. This was achieved by using the exterior angle property of a cyclic quadrilateral, as well as the interior angle property of a triangle.

References:

* NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 4

* NCERT Solutions for Class 9 Maths Chapter 10

Prove abcd is a parallelogram